How can I find words in a string that are related to a given word, then associate a sentiment to that found word?
There are many ways to solve this problem.  One way is to apply stemming or lemmatization to reduce your words. Using NLTK's Porter stemmer for example on healthy, healthier, healthiest, not healthy, more healthy, and zero healthy gives:
healthi , healthier , healthiest , not healthi , more healthi , zero healthi

This can help make word comparisons easier.
Sentiment analysis on the phrases will provide positive, neutral, and negative scores.  There are a lot of algorithms for doing this but a common one is Valence Aware Dictionary and sEntiment Reasoner (VADER).  Here is a recent article with code using NLTK and the VADER lexicon:

How To Run Sentiment Analysis In Python Using VADER by George Pipis

The following article also does sentiment analysis using NLTK and includes stemming and lemmatization.  Instead of VADER they use a Naive Bayes classifier on a labeled data set of tweets: How To Perform Sentiment Analysis in Python 3 Using the Natural Language Toolkit (NLTK) by Saumik Daityari.